The transcendence of e (part 3)

Recall that in step 1 and step 2 we proved that for any prime sufficiently large and that is a nonzero integer. In this step we will prove that if is large enough, then and hence cannot be a nonzero integer. This will give a contradiction and we will conclude that is transcendental.

Step 3.

Recall that . Expanding this using our polynomial

we obtain

Recall that and . So , , and . So we have

and (using the well-known fact that in the limit factorials grow faster than exponentials)

In particular, we can choose the prime large enough so that for , . So

as promised. This yields a contradiction and we conclude that is transcendental.